The Fuglede conjecture for convex domains is true in all dimensions

• DOI: 10.4310/acta.2022.v228.n2.a3
• Publication Date: 2022-07-01T16:40:43Z
• Authors: Nir Lev; Máté Matolcsi
• Keywords: Convex polytope

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. conjecture due Fuglede (1974) stated that $\Omega$ a and only it can tile by translations. While this was disproved for general sets, long been known convex body "tiling implies spectral" part in fact true. To contrary, "spectral tiling" direction bodies proved $\mathbb{R}^2$, also $\mathbb{R}^3$ under priori assumption polytope. In higher dimensions, remained completely open (even case when polytope) could not treated using previously developed techniques. paper we fully settle Fuglede's affirmatively all i.e. prove then polytope which introduce new technique, involving construction from crystallographic diffraction theory, allows us establish geometric "weak condition necessary spectral.